and \(\hat{R}(x) := \rho ^{-1/2}\hat{S}(x)\rho ^{1/2}\)  for all \(x \in \mathcal {U}(G)\)  defines the unitary antipode on \(\mathcal {U}(G)\) . See [1]} for these facts and the interpretation of functional calculus \(\rho ^{\zeta } = \exp (\zeta \log \rho )\) , \(\zeta \in \mathbb {C}\) . We have an action \(\vartheta ^{\zeta } := \mathrm {Ad}\rho ^{i\zeta } : \mathbb {C} \curvearrowright \mathcal {U}(G)\) . Since \(\rho ^{it}\)  falls in \(W^*(G)\) , it induces a flow \(\vartheta ^t : \mathbb {R} \curvearrowright W^*(G)\) , which is clearly continuous in the \(u\) -topology (see e.g., [2]} for the topology) and fixes elements of the center \(\mathcal {Z}(W^*(G))\) . Since the second formula in (REF ) leads to
\(\hat{R}(\rho ) = \rho ^{-1/2}\hat{S}(\rho )\rho ^{1/2} = \rho ^{-1},\) 
